Search: Lets you filter which layers to view (BG/FG independent). The X clears the search field, and A selects all layers.
+: Adds a new layer
-: Removes a layer
D: Duplicates a layer
R: Renames layer with options for replacing/trimming/prefix/postfix on multiple layers
Rs: Simple rename, just asks you for a new name on a single layer
I: Inverts the selection
O: Center Pivot. If multiple layers are selected, all layers will get their pivots centered
X: Cut a layers geometry and remember the layer name
C: Cut a layers geometry and remember the layer name
P: Paste cut or copied geometry into a layer and name it based on the original layer (if the geo was cut/copied via X or C)
UpArrow: Moves the Layer UP
DownArrow: Moves the Layer Down
M: Moves selected objects into a new Object
Eye: Defines the visibility of a Layer
Here's a how-to video showing OD Layer Panel Pro in action.